# ex-4-PCa.mod
# 
# Alternative Price Consistent NCP formulation of 
# Multi-Leader-Follower Game 
# 
# Example 4 from Fukushima and Pang, "Quasi-Variational Inequalities,
# Generalized Nash Equlibria, and Multi-Leader-Follower Games", to
# appear in Computational Management Science.
#
# NB: KKT1y and KKT2y are INCONSISTENT.
#
#######################################################################

option presolve 0;

# ... sets
set I := 1..2;

# ... primal variables
var x{I} >= 0, <= 1;         # ... leader 1 & 2
var y  >= 0;                 # ... follower
var s  >= 0;

# ... multipliers
var chi{I};
var psi;
var sigma;
var xi;
var mu;
var dummy; # needed to square system

subject to
	# ... first order conditions; player 1
	KKT1x:   0.5 - mu = chi[1];
	KKT1y:   1.0 - mu = psi - xi*s complements dummy;

	# ... first order conditions; player 2
	KKT2x: - 0.5 - mu = chi[2];
	KKT2y: - 1.0 - mu = psi - xi*s;

	# ... FO conditions wrt slacks s
	KKTs: mu - sigma + xi*y = 0;

	# ... definition of slacks
	DefS: s = -1 + x[1] + x[2] + y  complements  mu;

	# ... complementarity conditions player i
	C1{i in I}: 0 <= x[i] <= 1    complements  chi[i];
	C2:         0 <= y + sigma    complements  psi >= 0;
	C3:         0 <= s + psi      complements  sigma >= 0;
	C4:         0 <= sigma + psi  complements  xi    >= 0;